“A PROCEDURE FOR CHECKING
EQUALITY OF REGULAR EXPRESSIONS”
- A. GINZBURG
Presented by Kenneth Burgess
Ginzburg, A. "A Procedure for Checking Equality of Regular Expressions."
Journal of the ACM 14.2 (1967): 355-62. Web.
CONNECTION TO CLASS
FA -> TG -> RE -> FA
Polymorphism
 In acquiescence to the insatiable demands of a ravening
appetite, I require sustenance in the form of ingested
nutrition.
 Yes.
The Equality Dilemma
MOTIVE – WHY THIS MATTERS
Algorithmic
Which translates into…
FINALLY, THE PROCEDURE
Taking advantage of polymorphism
RE -> TG -> Finite set of left linear equations
Y = aYa + bYb + ^
Ya = aYaa + bYab
Yb = aYba + bYbb
If each possible pairing of derivatives share “^”, the
RE’s are equal.
THE THEORY BEHIND
Taking derivatives of a RE
Definition of a derivative
R = ab+bab(a+b)*
An example derivative: R(b) = ab(a+b)*
You try: R(a) =
LINEAR EQUATIONS
Now we take the finite set of !!unequal!!
derivatives
R = a + ba*
R = aRa+ bRb
Ra = aRaa + bRab + ^ = aRaa + bRab + ^
Rb = aRba+ bRbb + ^ = aRb + bRaa+ ^
Raa = aRaaa + bRaab = aRaa + bRaa
CHARACTERISTICS OF DERIVATIVES
Every RE has a finite number of unequal derivatives
 Similar to our proposition -> every RE will produce a
TG of finite states
 Every derivative has a corresponding set of
vertices in the graph
 In short, Rx is rep. by the same TG as R, but with Ax as
initial vertice
EXAMPLE – R
~
S
PAIRS
 # of distinct derivatives = # of distinct sets of beginning
inputs
 To consider equality, we have to consider all possible
beginning inputs for both regular expressions
 u (the number of pairs of inputs we have to consider)
<= mn
 u (distinct pairs of R and S) <= m (distinct derivatives
of R) n (distinct derivatives of S)
PAIRS OF R AND S
Pair (Rx,Sx)
Infers
(^,^)
(a,a)
(b,b)
(a,a)(b,b)
(ba,ba)
R has ^?
S has ^?
Yep
Yep
Yep
Yep
However, we do not continue, since ba is considered
equal to b
Since the columns R has ^? and S has ^? agree, the
two RE’s are equivalent.
TABLE REPRESENTATION
To streamline this process, we create tables of equivalencies – showing derivatives
(or inputs), vertices corresponding, and whether they include ^.
Inputs
^
a
b
ba
1
x
Vertices of R
2
Equal To
^?
b
x
x
x
3
x
x
x
HOW DOES THIS HELP US?
Examine T0
When 0 is inputted into the graph,
2 -> 4
3 -> 2
HOW DOES THIS HELP US?
Examine T^
When ^ is put into the graph,
what happens?
HOW DOES THIS HELP US?
A^ are the starting vertices
So A^ T^ = A^
Relational functions… so
2 T1 = 2 (does nothing)
2 T0 = 4 (input 2, get out 4)
2 T^ = 2,3
HOW DOES THIS HELP US?
- A(10) can be computed by relational
functions.
- Computers can produce the derivatives.
- Computers can check equivalency as long
as they keep track of which states are
accepting states
TO REVIEW
RE -> TG
Construct table representation of possible inputs
Represent TG as set of relational matrices
Determine U, the set of unique pairs of R and S
inputs
For each pair in U, determine if both R and S agree
on whether or not they accept